Continuous-Flow Graph Transportation Distances

Abstract: Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized approximations. Motivated by fluid flow-based transportation on $\mathbb{R}^n$, however, this paper introduces an alternative definition of optimal transportation between distributions over graph vertices. This new distance still satisfies the triangle inequality but has better scaling and a connection to continuous theories of transportation. It is constructed by adapting a Riemannian structure over probability distributions to the graph case, providing transportation distances as shortest-paths in probability space. After defining and analyzing theoretical properties of our new distance, we provide a time discretization as well as experiments verifying its effectiveness.